Drawing instrument.



lJ. W. WHALEN. DRAWING INSTRUMENT. APPLIQATION I'ILBD DBO. 1B. 1002.

ess l ..smnvwm v n u o 3 m Q R. .R3 3 om ,A d il v ,N T wQE; 8 ma 7 Y Ns 3.. NR? M. Hvhwu w J. W. WHALEN. DRAWING INSTRUMENT. MPMan-10N FILED1530.18, w02.y

latentea Nov. 5, 1912 `10 SHEETS-SHEET 2.

` W/ TNESSES: 9W Q.

J. W. WHALEN.

DRAWING INSTRUMENT.

lAPPLIMTION FILED Dnc. 1n Hmz.

1,043,789. rammed Nov.5,1912.

10 SHEETS-SHEET 3.

n v Wage/o OWMM 7L. W

l l HIV W 1%?? ArroH/VEYS.

J. w. WHALEN. DRAWING INSTRUMENT. APPLICATION FILED DBO. 1B, 1902.

Patented Npv. 5, 1912.

10 SHEETS-SHEET 4.

l @MQW y J. W. WHALEN.

DRAWNG INSTRUMENT.

APPLICATION FILED DBG. 1B, 1904. 1,043,789. Patented Nom, 1912.

J 10 EHEBTS-BHEET 5.

12E. JE.

W/TNESSES:

/NVENTOH /l/z yin/@Zen J. W. WHALEN.

DRAWING INSTRUMENT.

APPLIUATIDN FILED 13150.18. 1902.

1,043,789 Patented Nov. 5, 1912.

1o SHEETS-SHEET e.

9W 9 Z Wfl/75u21 n HY l l annul; *w'w' K TTHNEYS.

5J. W. WHALEN.

DRAWING INSTRUMENT. urL'IonIoN FILED 950.18. 1902.

Patented Nov. 5, `1912.

J. W. WHALEN.

DRAWING INSTRUMENT. APPLICATION r1LBI DG.18. 1902.

1,043,789. Patented Nov. 5, 1912.

l0 SHBETS-SHEET 8.

. A C/H l J. W. WHALEN.

DRAWING INSTRUMENT.

APPLICATION HLIJD n u.18.1so2.

Patented Nov. 5, 1912.

1o SHEETS-SHEET s.

M' n -A Macu/uf J. W. WHALEN. DRAWING INSTRUMENT.

APPLICATION FILED Dc.18.1902.

1,043,789, Patented Nov. 5, 1912.

10 SHEETS-SHEET 104- 1 .ofiansor .a

To all whom it mag/concern' Be it known thatfI, ,JOHN WHALEN, a

citizen ofthe nI lnited States, and a resident 'l'of-:Graymon n-tlilecounty of Livingston' 5and` State-:of Illinois, have invented a new M:aida -"1mpiro`.ved-;^l)tawving Instrument, of

`wvlii'clcaV the"nfolloyvingtis a, full, clear, and

exact description. l

The object of the invention is to provide l0 a new and improved drawinginstrument,

which is simple and durable in constru/ctcn,

easily manipulated, and arranged to permit of, first, measuring indegrees an are `of any circle whose dimensions are unknown and whosecenter is notar located; second, insuring accuracy in fthe`,measure-ment of angles and arcs; third,"`measuring line segments andradii and expressing theirlength in a certain exponent fourth, findingthe limitsl of each successivearc and the center of its respectivecircle in any curve, by means of which any curve may be inked with acoin pass; fifth, analyzing any curve'into measured arcs and'y radiifori-making a complete record; sixth,'using such recordY for reproducingthe segments ef curves or symmetrical curves.y of the same on anenlarged or reduced -,scal`c,;"fsev"enth, building complex "ni'i'dyeompound lcurves `and figurescomposed of straight lines and curves;eighth, forming designs composed of similar curves of varying sizes;ninth, reproducing a curvel in' a flattened, distorted or elongatedinanner; tenth, expressing the exponents of the algebraic expression(af) in which l) has the value of 40 eleventh, f'ineasuring distancesfrom or t0- ward a vanishing point in perspective drawff y,ingsby utheuse cfa certain scale; twelfth,

fLNiTED sTATEs PATENT oFFicE.

` Specification of Letters'illatent.

i" Nov 3;-

. vvlamination inea December is, iena.k serial Nq.,135,7e,i..

which similar characters of referencer indicate corresponding parts inall the views. Figure 1 isan enlarged face view4 of the drawinginstrument; Fig. 2 is a front edge view of the saine, part being insection; Fig. 3 is a transverse section of the same 60 on the line 3 3of Fig. l; Fig. 4 is a side elevationjof the needle point used inconnec. tion with the instrument shown in Fig. 1;

` Figs. 5 to 21, inclusive, illustrate various examples of the use ofthe'instrument; Fig. 6.

22 is an enlarged plan view of part of the improvement, showing thescale for fractions of a unit;l and Figs. 23,*and 24 illustrate examplesof. the use off' the scale for. fractions o a unit. The drawinginstrument shown in Fig. 1 is preferably in the f'orm of a 'rectangularplate A made of Celluloid, or other suitable transparent or translucentmaterial, and one of the faces of the said pla-te, preferably the lowerface, is provided with various working scales, numerals, etc., visiblefrom the top, and the plate A is also provided with sets of conical orcountersunk apertures adapted to be engaged by a'needle p oint B 80.

to allow marking centers, distances, etc., and

to permit of swinging the .plate A around with the needlepoint acting asa pivot, the plate bei-ng also provided with finger holes-1f C 1C', topermit conveniently lmanipulating the plate for the purpose hereinaftermore fully described.

Near the upper edge of-gplate A is ai'- ranged an apertured scale D,which reads from left to right from the starting point D near theleft-hand edge of plate A. The

.divisions of scale D equal in millimeters the algebraic expression ab,in'which algebraic ex ression a has the value of .78125 of a mi limeteror T12-8- of 100 millimeters; b has .95 the value of near theirrespective scale points. The smaller divisions, owing t-o theconflicting vco of apertures, cannot be given in regular order on thescale. The smaller divisions `1o l 2) i ""by the process of doubling andbisecting any number of graduation points may be \t`ound when tensuccessive Vpoint-s are given; for example, if it be desired to find 64(which is in nealityabei) vwhen 74 (or @574) is given, then 4 a-b ab,

and as 1o b1= was 2,

it follows that and may be found by dividing ab by two (2),r or bybisecting the distance D"-74 of scale D geometrically, 54 may be foundby dividing ab by four l(4) and 44 by dividing ab by eightx(8). In likemanner it -may be proven that 84 can be found by multiplying ab by two(2), and so on for any distance greater or smaller.

The ten divisions are as follows: ab7=100.000 millimeters. ab"1=107.178' ab72=114l869` ab78=123l14 m1213195() v@m2141421 @5762151571@H-:162.450 ab78:174.110

ab79=18606 y. In the scale D each division is the mean proportionalbetween its preceding and succeeding divisions, for example:

ab 1574 :y ab :c1175 ab.z3 ab IE7-FW By reducing the fractions In scaleDthe space between two succesabM-ablaL-ab and so on.

sive graduation points is themean proponi tional between the twoadyolningspacesf For example:

Now this does not hold true for anyvvalue of b except that used in thescale, namely, the tenthroot of two is proven below: required to provethat when 9 bwa; then abw-abHL-abse N o\wab 141.421 ab 131.950 alfa-ahw9.471

Now-

. 570 am :063 10's and- 16 ab and- 9.473=ab3 then LWL-ab (practically)=ab3.

Similarly itmay be proved that The quality of the scaleis of great valuein laying out distances .represented by figures which are less thanforty and which distances cannot be given in the regular order of thescales. Now as all graduations of the scale D are marked by apertures,they cannot' be placed closer to each other than 1 mm. without danger ofconfliction, for example, to indicate 17 or the distance D-17 on scale Din regular order would place an aperture between'. apertures sixteen andtwenty of the v scale,l which aperture would conflict with or break intoaperture 16. The distances which cannot be given in the regular order onthe scale are found fthe distance vbetween the l (55) and fifty-six (56)to equal the distances between consecutive oints that are givenon thescale, and the intervals between such consecutive points are indicatedby the numerals written upside' down, and which may be easily read andused by turning the scale around. In this manner every distancerepresented by the numerals from'one to thirty-seven (l to 37),

inclusive, may be found. 'To illustrate this quality of thescalefsuppose it be desired to c find the distance represented byseventeen (17,) which is not given in the regular position on the scale,but by turning the scale around seventeen (17 will-be found to equalpoints fifty-five There is no apparent reason why the dif ferencebetween any two successive divisions of the.scale D should equal someminor division of the same scale; but sincev it has been found thatalii-05674205536 it is evidenti that all other right side up numerals oflthe scale D bear the same relation tothe inverted numerals placedbetween them from the algebraic equation 11575,- ab ab ab'ls abn absa-The value of a in the algebraic expression ab isnot important, and if ais given'any other value the mathematicall efficiency of the scale'Dwill be the same. But since some of the measurements, which must beexpressed in the term` of ab yare quite small, it

is evident that the value of a must bemi-v nute. To conveywithsimplicity the ability of constructingthe'scale D, itis. necessary thatten successive divisions of'said scale begi'ven, and to dose.Withsuthcient accuracy requires that those ten divisions be large.

To secure the greatest simplicity in the mathematics involved inthecomputaionof those ten divisionsthe first one shouldcome out even orfree from fractions.

To satisfy the. above, practical require ments I give abw, or the firstof the ten d1- 'distances equal, respectively, t

drawn from their respec Vat points which limit dist 'scale D, whichdistance for each respective arcgraduatio .7 8125 mln, and 7) has th vif8 visions, the value of mme) one hundred l millimeters; and deduce:.theffvalu fof abby the following algebraic expression a .78125 min.The plate a is provided'wit apei the scale line of the scale'jDfto nserting the needle pointf'B i1 l scale line for convenient-1ya fdistances on paper, by punctiirin by a needle point through aperturesto'- obtain a. "'re points fty-eight (58) of the. scale D are drawn thecd E and E', respectively, 'n"aj o`w rection, as plainly indicated," 22.The centers of the arcs beginning point D- of thejscal arc E is dividedintoten parts points, the length of there the arc being such that'lin s,their respective graduat-iorrpo ts, dicular to scale D cut scaleDf'at,oint limit distances from Doffscla expressions abs?, cbm-Habib?, y y Aeach respective arc graduatiI,illWhiClralgf braio expression ahasvaluaof, .78125 mm. and b hasthe value o'f Vg,

VArc E is divided in to t w partsby graduation points, thelength th tiveparts of the arcfbeingx points, perpendicular to lsc le ,enses thealgebraic expression algebraic expression af, h,a s ,t else '/2 Linearscales F, F,'v` silfuilarj` ca lerD, are laid out along thebasejofftlxeiplat in Opposite directions franr protractor H is arranged@e p a to the center G. From the G as the center and graduated in half'degrees as y theunit beginning with zero at the intersect at` thepoin-t- J llocated a dist-ance beyond the. upper edge of the plate A,and

in t-he continuation ofthe zero line I-I of the prot-ractor, asplainlyindicated in the dotted lines in Fig. 1. From the point sixty (60) ofthe scales F, F are drawn arcs and radial lines similar to the arcs Iandl radial lines J extending toward the point J from sixtyi four(64)and similar arcs and radial lines are ldra-wn from every fourth point ofeach scale F, F down to the points sixteen '(16),'

it being expressly understood that all such radial lines run towardthepoint J. e, In addition to the scales, etc., already described, thefa'cefof the plate A is laid out with polygonalguresK of honeycombdesign, and preferably in the form o'f hexagons, each having one of itssides co-extensive .with aside ofthe adjacent hexagon, as plainlyindicated in Fig. 1. The plate A is also provided with a scale L runningalong at or near the middle of the plate, and this scale L is laid outl-in equal partswith an aperture 4at each divisional mark. -Adivi- :if-IA 'sional part L" of the scale at the left-hand end thereof issub-divided in ten twelve equal parts.

The left-hand portion of the plate A is and in vconsidered the positiveone and is marked vwith the plus sign ..-l-, and the right-hand portionof the said *plate is considered the negative side and r`isfprovidedwith the minus The device is used as follows: Were it desired to measurein degrees any angle of a circle whose dimensions are unknown and whosecenter is not located; for instance, to measure the arc al d of Fig. 5,when the center b and lines a b and c b are not given, then the operatorplaces the instrument over the arc al d so that the center G of theinstrument coincides with the point d and thrusting the needle pointvBthrough the aperture at G rotates the instrument about that'point untilsome one of the' arcs I is found `to coincide with the are d d aswhenthe instrument is in the primary position shown in Fig. 5. It will nowbe'noticed that the radial line G- 30on the positive sideof theinstrument run Io the point il, .Whichjra'dial line indicatesfthe'degrees of the are d d which, in example, is 30 degrees. Since4 allthe-arcs I are drawn tangent to the p erpendicular G I-I and whentheinstrument is in the position above stated, arc d d coincides withone of the arcs I, 4and thereforethe `erpendicular G H is tangent .to

the arc d d and theradial line 30 is evidentlyits chord; and since theangle formed by av vchord and tangent equals one-half the angle standingon the subtended arc, there- .fore angle I-I ,G 30 equals one-half ofthe angle a b c, but since the protractor H is laid out withone-halfdegreeas a unit, the radial line G 30 indicates thecorrectnumber of degrees for the arc d ci. If the 30 degrees arc d. d.had been a part of a larger or smaller circle', the operation ofmeasuring it would have been the same as that above de-g scribed and theradial line G 30'wouldhave` indicated its degrees. The point dof- `arcci d would have intersected the-'radialline G 801`at a point'respectively-farther from or closer to fthe center G as "rele was largeror smaller than'1 in the 'example illustrated.

Were it desired to locate the centerof the circle of'which the arc'iZ dis a part, the

operator observes what one of the lines J intersects with the arc Iwhich coincides with the arc d d and notes the intersecting point of theline J with the scale near' the base of the instrument, and which pointof intersection is the center sought, and as the radius of the circlelies inthe scale'- near the base of the instrument it may be measured byscale F. In order to measure distances and express the same in thelexponents of b and 'to increase or decrease the length of line segmentsor radii to any one o a number of ratios, I proceed as follows,reference being had to Flgs. 6, 7, 8 and 24,.;wl'rich latter figure moreparticularly illustrates fractional measurements hereinafter more fullydescribed. When the set of lines and radii a, 5,0, (land e, shown inFig. 6, is

'given and the lines are measured with either the scale D, F or F, theywill be found to index`40, 44, 49, 48 and 36, respectively. Now if it bedesired to increase the length `of line marked L to that of a in Fig. 7,and

to increase the lines 7)', 0,'0l a'nd e in like proportion, the operatormeasures the line a to find that it indexes 50, that is,`contains tenpoints more than the line a of Fig. 6. Now, the lines'b, 0, cl and e ofFig. 7 are laid ou-t by increasing each' of the corresponding-lines ofFig. 6 ten lpoints so that said lines in Fig.v .7 index 50, 54, 59, 58and .4G-respectively. lIf-.it Ibe desired to reduce .the line' a tothexllne r1.','-Fig. 8, and the lines, c, d' and e of Fig. Ginthe sameproportion, then .the line a is first measured with anyone of thescalesD, F, F and as 'many points (4) as'this line visshorter than theline a so many pointsv (4 l'are-taken yoff from wel; Of heflinesdbm, md.8, t0 PrO-- duce lines of correct proportionate length, as will bereadily understood by reference to Fig. 8.

' tracting for the reason that 'in reality adding o-n the scales isequivalent to multiplying by b to the power of the number used, and

subtracting is equivalent to dividing by the power of b.

l When it is desired to find the limits of arcs and the centers oftheirrespective circles in a curve or system of points in a curve, Iproceed as follows, special reference being had to Fig. 9. Vhen theellipse and its long axis, shown in Fig. 9, for instance, is given, theplate A is placed in position 1 With the center Gr at or and its basecoinciding with the long axis a--g- Now it Will be noticed that an areon the negative side of the instrument coincides With a portion of theellipse and a line connecting this arc with point i2 of scale F. Theneedle point B is now passed through the aperture 52, marking center No.1, and With this needle point as a pivot the plate is turned to positionll. or to a position Where the center Gr will start to leave theelliptic linev if the plate be pushed farther. Here the point D ismarked by the needle point B through the aperture G marking that point.Nowit Will be noticed that.another are on thev same side of theinstrument coincides With a portion of the ellipse, and a line connect-sthis arc with point 58 of'scale F. The needle point isfnovv inserted ataperture 5S, marking center No. 2, and with that as a center theinstrument is turned to position Hl., or toa point Where the center Gceases to follow the elliptic line. Here the needle point is inserted atGr to mark the point Now it Will be noticed that another arc coincidesWith al portion of the ellipse and that a line connects this arc withpoint 6st of the scale F. `This operation is repeated until the limitsa, b, c, (l, e and the centers of their respective circles l, 2, 3, 4,5' are found. With those limits of arcs and centers of circles theellipse may be inked with a compass. ny curve may in like manner bedivided into arcs of circles joined at their points of tangency, andwhen those limits of arcs and centers of their respective circles areonce found the curve may he worked With a compass. i

When it is desired to analyze, any curve into measured arcs and radii sothat a-complete record may betaken of the curve, the instrument is usedas follows, reference being had to Fig. 10. In order to analyze thespiral curve a, b, c, d, e, f', g the plate A is first placed over thecurve with the center G connecting with the point a, as indicated in thedotted lines, and then the needle point 'is insert-ed in the aperture ofthe center G and the plate turned until one of the arcs I coincides.\vith the curve a Z. This arc is connected by a line to a point 3Gon the negative scale F,`and With the point -36 as a center the plateisv turned the same as above described relative to Fig. 9, tosuccessively obtain the point 1, 2, 3, 4, 5 and 6. The

angle g 6 f or the arc g f is now measured vaccording to the rule laiddown above in the description relative to Fig. 5. and it Will be foundto measure 60 degrees and the 'center of its circle is on the point 56of the scale F,

Aso that the record for this are is 556-60, the

minus sign belonging to the are and not to the radius. By measuring theare j' e in a like manner it is found to be 60 degrees in length andhaving a radius of 52, and the record for the two arcs is Written Thenext arc c (l will be found to measure 60 degrees with a radius of 4S orfour points less than 52, and so on, and the entire vrecord will appearas shown in the said figure. N ow to reproduce the original curve whenthe record is given it is necessary to first place the plate on theblank drawing paper and then thrust the needle point 'B through thecenter G to leave a puncture on the paper. The needle point is nowinserted at point 56 on the. negative scale F as per record, and thenthe plate is turned until the puncture previously inadeappears beneaththe radial line (St-60, which is the second numeral on the record. `Thecenter Gr is now again marked on the paper by a puncture and with apoint- 4 spaces nearer center as a pivoting point for the plate, thelatter is again turned as before. This process is repeated until theseveral points g, e, d, c, b, a, and the centers 6, 5, 4,3, 2 and 1 arelaid out so that the curvemay be readily drawn in ink or pencil by theuse of a compass inserted successively at the centers te draw the arcs.To produce similar or symmetrical curves on the same or on a reduced orenlarged scale. it is only necessary `to follour the directions lastgiven but With the signs of the arcs reversed so that the curve startsfrom left to right, sce Fig. 11, instead of from right to left as shownin Fig. 10. By changing the left-hand igures l of the record of Fig.101e-66 and following the directions above given a larger but similarcurve, see Fig. 12, reproduced. By reducing the left-hand figure of .the

record in Fig. 10 to, say, 40 and proceeding as before, smaller butsimilar curve (see Fig. 13) ivll be the result. Now it is evident thatby changing the records corre@ spondingly to the range' of the'instrument reverse curves may be formed by changing the signs of someof the arcs.

'Io bui-ld complex and compound curves alid'iivures com osedvofstrawhtlines and curves, I proceed as follows. reference being had toFigs. 14, 15, 16 and 17: The curve shown in Fig. 14 is composed of onemain :spiral curve a, Z), c, cl, c and two branch curves b f g hi and cd', and may be called afcomplexcurve, and its record ls'indlcated "i ialongside the said figure. l This curve is laid outethe same as thecurve shown in Fig-12 and above described, with the exception that afterthe are b b has. been passed over l by the center G, which then lies atb, the

needle point is inserted at point 40 (12 less than the arc previouslydra-wn) on the positive scale F of the plate A. The limits of the arcsf, g, L, z' and centers 5, 6, 7, 8 are easily found by following thefirst branch ot' the record. In this manner any number of branches maybeadded at pointsnindicated.' The record and branches may be provided withadditional sub-branches.

In Fig. 15 is shown a compound curve or the combination of two maincurves a b and b c Z joined by a sharp angle at In laying out Fig. 15,the needle point B is inserted at the apertured center G, leaving. apuncture at a, the needle pointA is then inserted at 40 of the negativescale F', making a puncture at 1, and about that point the instrument isrotated in a negative or anti-4 clockwise direction until the line G 90comes in contact with the puncture made at a. A,The -needle point isthen inserted at G leavinga puncture .at a, and the instrumentzisjotatedas before about point 1 until the `line G 90 comes in contact wit-h a.With-fthe instrument in this position Athe needlf. point is inserted atH leaving a puncture at it. The needle point is then inserted at Gleaving a puncture at and about this point the instrument is rotated iny a 'positive or clockwise direction a distance of 90, that is, thepuncture made at L is made to appear as if it passed beneath 180 degrees`of the protractor H 'graduated with twice the correct number of degreesas pre* viously mentioned. Th'e"needle,point B is neXt inserted at 50,orllQ'more than the last 'dimensi'on used o'n the negative scale F', andwith this point as acenter the arc b c is laid out; Y

The figure shown in Fig. 16 consists of lthe combination of two straightlines b a and a c and the curve c d e j". To l-ay out t'hls combinationof lines the center G .of

plate Ais placed at point b and point a is located by passing theneedle4 point B through the aperture48 of the scale F. The

`center G ot the instrumentisthen made to coincide withv point a andscale F is made to coincide with the line ba; theI needle point is theninserted at H leaving a puncture at It, andthe instrument is Vmade torotate ninety degrees ina positive or clock wise direction about pointa, causing the 4protractor 'H to pass one hundred and eighty of its halfdegree units over the puncture made at h, point c is thenylocated bypassing the needle point B through aperture 86 (or 12 spaces4 less than18) of' r" Scale F; thecenter G is then made to coincide with point cand scale F made to coin'A cide with. the line a c. They curve c d c fis Fig. 17 represents designs built up by the combination ofico'mpleicurves and in each successive case'flthel positive and negative signs ofthc arcs are reversed and the left# hand upper figure of the record isreduced by 4. It will bereadily seen that a large number of designs maybe made by the combination of similar complex curves,by simply modifyingor changing the left-hand upper figures and t-he signs of the arcs inthe record. Owing to the fact that compound curves and combinations oflines can be made to undergo the'same changes in relation to size it isevident'that the number of different designs made 'by their combinationis practically' without limit. By the adop-l tion of a few simple rulesthe curved lines are kept Within the bounds of grace and beauty.

In lorder to elongate, distort or flatten curves in a uniform manner,the following is to be observed: Itwill be noticed that the ellipseshown in Fig. 9 is simply a flattened circle 'composedoftwelve 30 arcs,

the radii of the same'varying by steppings of (Espaces or points on thescale of the instrument. The record of this ellipseis shown in the saidfigure, and from thisrecord it will be'seen that the radius of theoriginal circle is practically 58 points..

The ellipse of Fig. 9 may be develo d froma circle in the followingmanner:

and to each successive left-hand portion of this record add respectively(-6). (+6),

and-the record of the ellipse is obtained.4

by the use of this instrument and without, losing the smoothness orcharacter or theV curve.

In measuring angles slight errors may arise either from inaccuracy .inthe direction of the instrument, or inaccuracy in handling it. In layi/gout long continuous curves or spirals an/ error may add to another errorand thus the -figure becomes very inaccurate. In order to prevent theseerrors use is made of the hexagon figures K, in the following manner:Light lines, preferably in pencil, are drawn over a sheet of paperparallel to the base 'of the instrument in its first position. with thelines less than the width of the instrumentapart, and after theinstrument is rotated'labout any of its -,\.oint4 through the arc lifOdegrees.y it is ,identi that the lines joiningthe centers oi thetarragona wim in wave weinig an -grees or every multiple of Qjdegi-eesiangles employed are multipl enlarged or reduced -in 'sizeiandmadetryin?-A `Fig. 19, and as the distancef/t angle of 30 `degrees with thebase, will come parallel to the lines ongthejpaperL-f; YVlien rotated 30degrees more,., or sixty ,degree,s-t.- the light lines on the papervwill be Iparallel e to two sides of every hexagong; andL-,whenro-m tated30 degrees 'farther,ftgr ,ninety g degre then the lines becomeparallel;l tegthos that connectthe centers of the henago; in rowsperpendicular togthe base accrued error may be corrected e, V

arcs of 30 degrees or anygultiple offO, de-

14, 15, 1G and 17, are t0 beleid Out-,ther their@ s; hexagoneA may beused asofumeasfgg urement, Confusion mayresul't in 'that case.

however, as no figures would :befindicated. r

but by "depending on the:protractorA H for the indication 'ofdegrees,.andion, ,the hexagc-ns for accuracy, it, isevidentg; hat?e'eiymeasurement in degrees becomesp crivcally, perfect, for the specialcaseyglin; vhic grecs (30), v .mr-i Y The protractor fis aynelfythroughout its'entire length,v sthA ty notbnly; a very Vetficientprotractongzisrgprvided, ,bv it permits using the instrumentformt-nanterring a system of points ,to the exactlocaf tion desired and thesystemwof-points mayxbe; `i951. I

pear in the same 4or symmetricalglfortn.lille illust-rate this methodbrietlytxlrlpnoced follows, special referenceibeingihalrtoFig A, 18 and19.` If the systemsolzpointsrd, b,- f e f g z' of thecurve-,shownicinali- .f1 is given and it be requiredftb tnansferrandenlarge this given system o fpoints andi make it appear in both the same"and symmetrical, form to produce the capyonvt-he ,tpaiitwliislc shownin Fig. 19, then the ,nstrumetisfnsed @ml tive tothe pointsa c.,.,andthentith-roug in any desired direction'fategan pointirel tive tothe points a. b o .,1andfclrensitlrrouglr ,5-z the point r: is drawn a'line perpendicular' to a0 the line s 0 tointersect this li-ne'at'thecpoint ig; Z and at some pointy in the'iline ssffprefe g ably atVor near the center of thepurtf irr que tivi tion, is locatedr the pointh.v` frNotx suppose. e ai. it hasbeen determined to doublethe. size d 15 lthis system of points, or in. terinstoffthis in?? .st-rumeni',increase each dimensioni;fv f The center line s o 1s lald outfas 1ndequals according t0 the scale Ff, it is evident that the 001:56 1.

63%- points, of the scale DQKFQF v point Z. The center G offtli'e:prow-actor' H is now placed over the poirrtjlfitltjgsit L. and theneedle point B inserte ttthereehtenif aperture, and then the instrumentrotated 13o until the positive scale F passes over the point a. y

Now it will bel found that the point a.' lies between the aperturesnumbered 50 and 51,

- and the line s coincides with 147 on the positive side of theprotractor H and the record for the point a 1s therefore +147 -50;

the instrument is then rotated about" the point It until. the positivescale F. passes the point Z), which coincides with the aperture numbered51 on the scale, While the line s 0 lies under 105.011 the same side ofthe prot-ra-ctor, and the record for the point l) is written -|-105-51.This operation is repeated until the entire record is found, the saidrecord being preferably Written as indicated in Fig. 18. Now, in orderto reproduce this system of points on an enlarged scale as indicated inFig. 19,v each linear distance 1s increased, say by ten, on the scaleI), F or F', and the record 1s made to read asindicated to the left ofFig. 19. The

center G of the protractor is now placed at `the point la, (Fig. 19) andthenythe instrument is rotated about that point until 147 .on thepositive side of the protractor coinv,and the two punctures associated"with it.

In a like manner all of the point's-i'a b c d e f g h i j lo arelocated. To produce th'is system of points in symmetrical form` or tolocate points a b c shown in Fig. 19, the positive and negative signs ofthe angular record are first reversed so that the angular record readsas indicated to the right ofFig The operator now proceeds the same as.hen laying out pointa b c only using the negative side of theprotractorscale :instead of the positive. The system of points'may be transferred,by this method, without marking a record or noting the degrees. Toillustrate this the small curve a2 b2 c2 shown in Fig. 18, is given. Thecenter G of the protractor is placed at the point h, and then theinstrument is rotated .about-this point until eitherl the scale F or Fpasses over the point a, and then the needle point B is passed throughthe apertures numbered 50 and 51, to form4 punctures on .both sides ofthe pointa. New supposing that a curve is to 'be reduced by bs; Whilebeing transferred it is necessary to pass the needle point B through theperfol`rations numbered 42 and 43 of the scale on the opposite side ofthe instrument, and the .positionof the point a between the punctaresyformed Vis estimated, with the eye.

' tioned, the following will appear similarin "shape, 4or upside downand reduced or enlarged as the case may require iVhenthis process is'used to transfer drawings lfrom one sheet' to an other, then the line so should constitute the edge of the sheet on which the original drawinga b c .exists and also the edge of the sheet tc which the drawing istransferred. The points g i j c of Fig. 18 are laid out similar to thecorresponding points of the curve shown in Fig. 18, with the exceptionthat only one scale 0f the instrument is used during the operation. Theprocess is useful in perspective drawing. In a number of theabove-mentioned cases the exact location of, the point to be found mustbe estimated;

exact location of the point in question in all To illustrate theseincommensurable cases. the use of the mechanical means just menis given:the example being transferring t e point g in Fig. 18 to tothe point gin Fig. 18. The instrument is first rotated about the point It, asbefore dci scribed, until the scalel F extends over the point g. Theneedle point B is now passed through the first aperture lying toward hfrom g to form a puncture, at the same time passingthe needle pointthrough the aperture numbered 55 to leave a puncture in the vicinity ofg. A perpendicular line is now drawn from the point g to the lineso',..,and then the instrumentis rotated about' the point la until theaperture numbered 50 comes in contact with the perpendieii'lar line justdrawn, and with the instrument in this position the needle pointB ispassed through the aperture numbered 55, and through the puncture formed'a perpendicular line is drawn to the line s o and the point ofintersection of this perpendicular, and the line 71.55 oi' h g, producedis the proper location When producing a record.-

of the point g. which involves this means of locating point. twopositions of the instrument must- Y,

be indicated ;vtliat is, When-the instrument is 'rotated about the point71, until the scale F comes in contact with the point g, then the line so coincides with the negative 80". and when the instrument is rotatedabout the point l1, until the aperture 50 comes in coutact with theperpendicular drauf'n from y] to s o, then the line. s o coincides withl he negative 83 and the record for 'the point q is read *800 and7832-50. true for all incommensurable cases.

.The scale D is also valuable in making perspective drawings, Iand toassist in this kind of work I provide the instrument with the scale L ofequal parts. In order to more fully bring out Vthis feature, I proceedas fol lThis holds lows, special reference being had to Fig. 20,

which represents a pile of cubes upon an area six (6) units long andtive (5) units Wide; over the right half of this surface the cubes arelaid two in height and over the a left half they are four cubes inheight, and the figure vis produced asfollows:4

The line m a9v representing the horizont-al is first drawn and then t-heinstrument is placed in the position shown in Fig. that is, the startingpoint. D of the scale D is i -caused to coincide with the point m2 onthe horizontal line .fr m so that the scale D stands at right angles tothe said line 'The needle point B is now passed through the aperturenumbered 60 ofthe scale D, and then a line parallel to 0aY is drawnthrough the puncture made say at 60, and this line is considered vthebase line for the figure, and all distances to the right and left arelaid out on this base line, While distances in height are laid out onthe perpendicular lines drawn from this base line. It is understood thatthis base line may be produced fron any of the apertures in the scale DWith the result of making the object appear at a greater distance, orrather With t-he result of changing the point ot' observation of theonlooker relative to the figure. On the base line are laid out theequi-spaced points t b in the said scale. Now from the points a ZJ cperpendicular lines are constructed which are two units in heightaccording to the scale L, and from the points cl e j g perpendicularlines `four units in height are drawn, and then the lines g4 d4, g3 d3,g2 0,2 and g a are drawn parallel to the baseline a g. After this hasbeen done the radiating lines m2 29,5122 a,2 a, m2 b2, m2 c2, m2 d2, x2cl3, as? d4, et, m2 f4 and fm2 g4 are drawn. The instrument is nowplaced on the igurein such a manner that the point D of `the scale Dcoincides with the vanishing point m2, and' -then the instruinent isrotated about this point until some point of the scale D, say

57, comes in contact With ther line a, a2; then i. the operator passesthe needle point through until `Isome ypoint of the scale D, say 61,cornes in contactvvith the line a2 vrZZyand then'the operator passes theneedle ypoint through the aperture 61 .andthrough the. five precedingapcrtures/ as indicated ini. Fig. 20; and through each of the puncturesmade, lines parallel to a (Z2 are drawn limited by m2 a2 and m2 (Z2.rotated about the point m2 until some point The instrument is'now againm2 (57), divides the line nu of which parts the part conn equals J Withm2 equals w A* .in

d li-1th n' t ."a'bQVeam parallel to a2 d2 and pas l in line m2 6ldivides the parts, of which parts,` at with ma equals i, if

v:71;256 =ab5"=-L,-b-

and since it is a `,geolletrica l parallel lines divide alll linesfraing from ,.1 the same point into liketprdprtions, theliner which isparallel to a? hand-.passes througlu para and passes through .pointef'alii which .is parallel to linehrify i. an'd" lthrough point 56,cuts'ithefinef u points Which areA identical die mannen-' it may beproved that eve which meets a2 andmit perpendicular line which meets theline w? a2 at points which In like manner it may bel rovetlbthat-comfeling point are obtained from the sub-divisions Fig. 2l represents` a.

and the'p'olvit r is located units l to th-*right on the same line byuse `of the 'sca I mentioned, it being understood that`" the iding unitportion L of the said scale mployed to obtain the fractions. From thepoints 1' seven units in height, to obtain thepoints s1, 7'", which arevconnected with each other by a straight line. On the line s s1 are laidoli the point'ss4 and s55 at distances re- 2`5 spectively four and fiveand `a lhalf vunits l from the point s, and then the radiating z. lines:v2 sT m2 s55, m2 s4, :v2 s and m21. 7' are *di-Jaw .T e startingpoint Dof the scale Di' now placed on the point ma in the hori` zontal line aim', and then the instrument is rotatedaboutthat pointuntil the scale Dv"sti'iii(legi-tf-'right angles to the said horizontal line', and thenthe operator passes the needle point through the aperture numbered 59,

, countei "Jockwise about the point vuntil the divi; in 6 of the scale Ecoincides with the puncture just made, and with the instrument in thisposition the` needle point is` through `the "aperture liil'imbered 57,and

- through the puncturei'na'de thus and numsberedla'line paralleltothel'hrizbntal linea is drawn whichcu'ts the line :v2 's at u'.

' ThiS point u isl threeunits to the-rear of e' and by rojecting'r'omthe point u the tlaoint until the division dof-the scale E coincideswith the puncture*"59,' and 4then "lsthe'needle point is passedthroughvthe aper- 65 ti're numbered 6 2-ofthe'scalei1) so as to llel to thehorizontal l1ne w auf,

and s perpendicular lines are constructed,

after which the instrument is rotated tio the horizontal line :c isdrawn-and g e left 'armof the crossis formed. llhe instrument is l"nowagain rotated about leavethe puncturel praetif" Illy' 2,6f one andone-half units clOgL '-thBi-.Qbsrverl than the line s r6", through liincture a horizontal 'line' 'is drawn. 1 tit e' inter.-

sections of the line' just niadlwitlithe. lines l m2 s and w? nextended,perpendicular lines are drawn, andat the .intersections-ortheperpendicular line drawn fronifthe point v with the lines :v2 a*extended and w? '-s exf tended, parallel Alines are drawn'to the ri lit,and limited by the y perpendicular ine drawn from rw', These parallel.lines with the segments which they ycut oif--from the perpendicularlines drawn from 'v and w form the'boundary of the right arm ofthecross. The desdripti'oiiiof Fi e handling of the instrument 'when ,1%ofbt vunit used are involved. L" 'It'vwill'be notlced g.illustratesite'. e

that when the scale D has been rotated aboutA wa then the aperture ofthescale Dnears -the line :v ai', and when the aperture 60 comes asclose" the line `:v w* as thea erture marked 59 vwhen the scale isperpen icular,

then the distancew m 59:58 on'the scale' when perpendicular," and w w58:57 on the scale when perpendicular.A l And had f5, A125, it; of theunit used beeninvolved, then the first, second, third, and soon,'division of the arc E. drawn from 59 wouldl have lbeen used. ThisarcE' enables 'aiinit'close or lfar from the'Y vanishing point to bedivided into halves, thirds, fourths, sixths and twelfths' by usingrespectively 152, g, 1%, fig' 'and 115, In like manner the are whichextends from the aperture ,58jenables each unit of distances from ortoward the vanish*- ing pointwz to be divided into halves, fifths ScaleE or arc'E is so drawn romits dicular to scale graduated that lines.

which equal respectively valii-7, abs-, 6W- 2,1 ab57.8 n

ab581r, abf', e By' the combined 'ico and tenths by using respectively156, 126, :115;

or using the divisions 5, .2 and l4 of thlo graduatlon points',v1perpgn-' limit distances from 'D'. lief use of vthese two'aircs 'i 'l`and E anyv unit of distance toward orfioiti 4the vanishing 15911111 m2may be` divided into. halves,'thirds, fourths, fifths, sixths, tenthsand twelfths.' All the perpendicular and horizontalfilistances aremeasured by the 'scaleof equal parts L, and the measurements are takenin thelplanefwhich contains 'the points r, s, ahw", and all the pointsin the vplanes which are closer or farther from the p lane mentioned arefound by the intersection of lines; and all distancestoward thevanishing point arefirst laid out on the ioor orthe plane which containsthe points'

